# Reversibility of a Markov Chain

Rephrased question: Is it ever possible for a reducible Markov chain to be reversible?

-

Reversibility implies that if the Markov chain can go from $x$ to $y$ in finite time with positive probability, the same holds for $y$ and $x$. Hence the only way to be reducible is that there exists some states $x$ and $y$ such that the chain cannot go from $x$ to $y$ neither from $y$ to $x$. In other words there exists a partition of the state space such that the Markov chain starting from a state in a given class stays in this class forever with full probability and such that the Markov chain restricted to this class is irreducible.
In other words, one considers a collection $(Q_i)_i$ of reversible irreducible transition kernels on disjoint non empty state spaces $(S_i)_i$ with stationary measures $\mu_i$. The state space of the Markov chain is the union of the spaces $S_i$ and while in $S_i$, the chain uses the kernel $Q_i$. Every barycenter of the stationary measures $(\mu_i)_i$ is stationary and the chain is reversible but reducible as soon as there is more than one state space $S_i$.